Dynamic Foraging Patterns in Antarctic Humpbacks
Ben Weinstein
2017-05-20 08:50:34
## # A tibble: 2 × 2
## Animal `max(timestamp, na.rm = T)`
## <int> <dttm>
## 1 123232 2013-09-28 07:28:00
## 2 123236 2013-03-18 11:26:00
0.1 By Month
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
1.3.1 After filitering
How did the filter change the extent of tracks?
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha[state[i,g,t-1],Month[i,g,t]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t],Month[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha[state[i,g,steps[i,g]-1],Month[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
#for each month
for (m in 1:Months){
#Intercepts
logit(alpha[1,m]) <- alpha_mu_invlogit[1,m]
alpha_mu_invlogit[1,m] ~ dnorm(alpha_mu[1],alpha_tau[1])
logit(alpha[2,m]) <- alpha_mu_invlogit[2,m]
alpha_mu_invlogit[2,m] ~ dnorm(alpha_mu[2],alpha_tau[2])
gamma[1,m] ~ dnorm(gamma_mu,gamma_tau) ## gamma for state 1
dev[m] ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2,m] <- gamma[1,m] * dev[m]
}
##Behavioral States
#Hierarchical structure across months
#Switching among states in inv.logit space
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)
#Variance in state change per month
alpha_tau[1] ~ dgamma(0.0001,0.0001)
alpha_tau[2] ~ dgamma(0.0001,0.0001)
#spatial autocorrelation prior, we know that it has higher autocorrelation than state 2
gamma_mu ~ dnorm(0.8,500)
#variance in gamma per month
gamma_tau ~ dgamma(0.0001,0.0001)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)sink()
## user system elapsed
## 3.14 1.36 312.331.4 Chains
## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1338933 71.6 2637877 140.9 2637877 140.9
## Vcells 10281792 78.5 28941810 220.9 46990700 358.6## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1331479 71.2 2637877 140.9 2637877 140.9
## Vcells 6799917 51.9 23153448 176.7 46990700 358.6
1.5 Change in autocorrelation over time
2 Change in transition probabilities over time
2.1 Parameter Summary
## parameter par mean lower upper
## 1 alpha alpha[1,1] 0.65211986 5.404306e-01 0.8517767
## 2 alpha alpha[2,1] 0.61082579 1.465187e-01 0.9999966
## 3 alpha alpha[1,2] 0.67527356 5.657484e-01 0.8526467
## 4 alpha alpha[2,2] 0.14329323 1.560355e-05 0.2967653
## 5 alpha alpha[1,3] 0.63383094 4.394857e-01 0.8348255
## 6 alpha alpha[2,3] 0.15557480 1.468828e-02 0.2824026
## 7 alpha alpha[1,4] 0.65161435 5.454670e-01 0.8181779
## 8 alpha alpha[2,4] 0.13547877 7.162426e-08 0.3033733
## 9 alpha_mu alpha_mu[1] 0.66103027 1.965716e-01 1.5889844
## 10 alpha_mu alpha_mu[2] -0.70449681 -1.938750e+00 1.7254560
## 11 alpha_tau alpha_tau[1] 1543.05694105 1.375895e+00 9363.6339378
## 12 alpha_tau alpha_tau[2] 1035.99819039 5.499725e-03 6628.2267366
## 13 gamma gamma[1,1] 0.36460916 -1.519343e-01 0.7647826
## 14 gamma gamma[2,1] 0.17093458 -7.931959e-02 0.4782040
## 15 gamma gamma[1,2] 0.01420102 -9.308402e-01 0.8699413
## 16 gamma gamma[2,2] -0.21302089 -6.213377e-01 0.2545191
## 17 gamma gamma[1,3] 0.68965269 4.821787e-01 0.9611908
## 18 gamma gamma[2,3] 0.21915097 4.543892e-02 0.4227511
## 19 gamma gamma[1,4] 0.38866293 -4.070012e-01 1.1727539
## 20 gamma gamma[2,4] 0.12357352 -1.453662e-01 0.4255142
## 21 gamma_mu gamma_mu 0.77990701 7.047316e-01 0.8667016
## 22 gamma_tau gamma_tau 91.50130912 2.730430e-01 539.6248497
## 23 theta theta[1] 0.06008947 -1.086422e-01 0.1956118
## 24 theta theta[2] 3.13793313 2.706122e+00 3.4130639
3 Behavioral Prediction
3.1 Spatial Prediction
3.2 By individual
3.3 Autocorrelation in behavior
3.3.1 As single timeline
4 Simulated tracks
4.1 Behavioral description
4.1.1 Proportion of states by month
4.1.2 Distance between bouts
5 Behavior duration
## Source: local data frame [7 x 7]
## Groups: Animal, Track, Bout, phistate [7]
##
## Animal Track Bout phistate MonthF Days Month
## <dbl> <dbl> <int> <fctr> <dbl> <dbl> <fctr>
## 1 4 1 1 Traveling 1 22.9841898 January
## 2 3 1 1 Traveling 1 18.8980208 January
## 3 2 1 4 Traveling 3 3.4944444 March
## 4 1 1 3 Traveling 3 0.9847222 March
## 5 1 1 5 Traveling 4 0.4659722 April
## 6 1 1 1 Traveling 2 0.3548611 February
## 7 2 1 2 Traveling 3 0.3215278 March## Source: local data frame [12 x 7]
## Groups: Animal, Track, Bout, phistate [12]
##
## Animal Track Bout phistate MonthF Days Month
## <dbl> <dbl> <int> <fctr> <dbl> <dbl> <fctr>
## 1 1 1 2 Area-restricted Search 2 40.2013889 February
## 2 2 1 1 Area-restricted Search 2 23.3729167 February
## 3 4 1 2 Area-restricted Search 2 23.0990046 February
## 4 1 1 6 Area-restricted Search 4 22.2701389 April
## 5 4 2 1 Area-restricted Search 2 18.9131829 February
## 6 3 1 2 Area-restricted Search 2 17.2529514 February
## 7 1 1 4 Area-restricted Search 3 10.9611111 March
## 8 2 1 3 Area-restricted Search 3 5.8590278 March
## 9 2 1 5 Area-restricted Search 3 4.4194444 March
## 10 4 4 1 Area-restricted Search 3 1.3696991 March
## 11 4 5 1 Area-restricted Search 3 1.0939931 March
## 12 4 3 1 Area-restricted Search 3 0.6460301 March## Source: local data frame [19 x 7]
## Groups: Animal, Track, Bout, phistate [19]
##
## Animal Track Bout phistate MonthF Days Month
## <dbl> <dbl> <int> <fctr> <dbl> <dbl> <fctr>
## 1 1 1 1 Traveling 2 0.3548611 February
## 2 1 1 2 Area-restricted Search 2 40.2013889 February
## 3 1 1 3 Traveling 3 0.9847222 March
## 4 1 1 4 Area-restricted Search 3 10.9611111 March
## 5 1 1 5 Traveling 4 0.4659722 April
## 6 1 1 6 Area-restricted Search 4 22.2701389 April
## 7 2 1 1 Area-restricted Search 2 23.3729167 February
## 8 2 1 2 Traveling 3 0.3215278 March
## 9 2 1 3 Area-restricted Search 3 5.8590278 March
## 10 2 1 4 Traveling 3 3.4944444 March
## 11 2 1 5 Area-restricted Search 3 4.4194444 March
## 12 3 1 1 Traveling 1 18.8980208 January
## 13 3 1 2 Area-restricted Search 2 17.2529514 February
## 14 4 1 1 Traveling 1 22.9841898 January
## 15 4 1 2 Area-restricted Search 2 23.0990046 February
## 16 4 2 1 Area-restricted Search 2 18.9131829 February
## 17 4 3 1 Area-restricted Search 3 0.6460301 March
## 18 4 4 1 Area-restricted Search 3 1.3696991 March
## 19 4 5 1 Area-restricted Search 3 1.0939931 March
5.1 Proportion of time allocation
## Month Traveling Area-restricted Search PropF TotalTime
## 1 January 41.8822106 0.00000 0.0000000 41.88221
## 2 February 0.3548611 122.83944 0.9971195 123.19431
## 3 March 4.8006944 24.34931 0.8353107 29.15000
## 4 April 0.4659722 22.27014 0.9795052 22.736115.2 Number of bouts
6 Time spent in grid cell
6.1 All years
6.2 Add in sea ice
